{"paper":{"title":"Local and global sharp gradient estimates for weighted $p$-harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nguyen Duy Dat, Nguyen Thac Dung","submitted_at":"2015-05-28T10:11:30Z","abstract_excerpt":"Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $\\lambda_{1,p}$ on $M$, namely $$ e^{f}div(e^{-f}|\\nabla v|^{p-2}\\nabla v)=-\\lambda_{1,p}v^{p-1}.$$ in the distribution sense. We first give a local gradient estimate for $v$ provided the $m$-dimmensional Bakry-\\'Emery curvature $Ric_f^{m}$ bounded from below. Consequently, we show that when $Ric_f^m\\geq0$ then $v$ is constant if $v$ is of sublinear growth. At the same time, we prove a Harnack inequality for weighted $p$-harmonic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07623","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}